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3.3: The First Derivative Test

Learning Objectives

A student will be able to:

Find intervals where a function is increasing and decreasing.

Apply the First Derivative Test to find extrema and sketch graphs.

Introduction

In this lesson we will discuss increasing and decreasing properties of functions, and introduce a method with which to study these phenomena, the First Derivative Test. This method will enable us to identify precisely the intervals where a function is either increasing or decreasing, and also help us to sketch the graph. Note on notation: The symbol and are equivalent and denote that a particular element is contained within a particular set.

Definition

A function is said to be increasing on contained in the domain of if whenever for all A function is said to be decreasing on contained in the domain of if whenever for all

If whenever for all then we say that is strictly increasing on If whenever for all then we say that is strictly decreasing on

We saw several examples in the Lesson on Extreme and the Mean Value Theorem of functions that had these properties.

Example 1:

The function is strictly increasing on :

Example 2:

The function indicated here is strictly increasing on and and strictly decreasing on and

We can now state the theorems that relate derivatives of functions to the increasing/decreasing properties of functions.

Theorem: If is continuous on interval then:

If for every then is strictly increasing in

If for every then is strictly decreasing in

Proof: We will prove the first statement. A similar method can be used to prove the second statement and is left as an exercise to the student.

Consider with By the Mean Value Theorem, there exists such that

By assumption, for every ; hence Also, note that

Hence and

We can observe the consequences of this theorem by observing the tangent lines of the following graph. Note the tangent lines to the graph, one in each of the intervals

Note first that we have a relative maximum at and a relative minimum at The slopes of the tangent lines change from positive for to negative for and then back to positive for . From this we example infer the following theorem:

First Derivative Test

Suppose that is a continuous function and that is a critical value of Then:

If changes from positive to negative at then has a local maximum at

If changes from negative to positive at then has a local minimum at

If does not change sign at then has neither a local maximum nor minimum at

Proof of these three conclusions is left to the reader.

Example 3:

Our previous example showed a graph that had both a local maximum and minimum. Let’s reconsider and observe the graph around What happens to the first derivative near this value?

We observe that the tangent lines to the graph are positive on both sides of . The first derivative test () verifies this fact, and that the slopes of the tangent line are positive for all nonzero . Although , and so has a critical value at , the third part of the First Derivative Test tells us that the failure of to change sign at means that has neither a local minimum nor a local maximum at .

Example 4:

Let's consider the function and observe the graph around What happens to the first derivative near this value?

We observe that the slopes of the tangent lines to the graph change from negative to positive at The first derivative test verifies this fact. Note that the slopes of the tangent lines to the graph are negative for and positive for

Lesson Summary

We found intervals where a function is increasing and decreasing.

We applied the First Derivative Test to find extrema and sketch graphs.